Improved Hardness of Approximation for Geometric Bin Packing

The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing where the input is a set of $d$-dimensional rectangles, and the goal is to pack them into unit $d$-dimensional cubes efficiently. It is NP-Hard to obtain a PTAS for the problem, even when $d=2$. For general $d$, the best known approximation algorithm has an approximation guarantee exponential in $d$, while the best hardness of approximation is still a small constant inapproximability from the case when $d=2$. In this paper, we show that the problem cannot be approximated within $d^{1-\epsilon}$ factor unless NP=ZPP. Recently, $d$-dimensional Vector Bin Packing, a closely related problem to the GBP, was shown to be hard to approximate within $\Omega(\log d)$ when $d$ is a fixed constant, using a notion of Packing Dimension of set families. In this paper, we introduce a geometric analog of it, the Geometric Packing Dimension of set families. While we fall short of obtaining similar inapproximability results for the Geometric Bin Packing problem when $d$ is fixed, we prove a couple of key properties of the Geometric Packing Dimension that highlight the difference between Geometric Packing Dimension and Packing Dimension.Comment: 10 page

Lower bounds for several online variants of bin packing

We consider several previously studied online variants of bin packing and prove new and improved lower bounds on the asymptotic competitive ratios for them. For that, we use a method of fully adaptive constructions. In particular, we improve the lower bound for the asymptotic competitive ratio of online square packing significantly, raising it from roughly 1.68 to above 1.75.Comment: WAOA 201

Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing

We study the generalized multidimensional bin packing problem (GVBP) that generalizes both geometric packing and vector packing. Here, we are given $n$ rectangular items where the $i^{\textrm{th}}$ item has width $w(i)$, height $h(i)$, and $d$ nonnegative weights $v_1(i), v_2(i), \ldots, v_{d}(i)$. Our goal is to get an axis-parallel non-overlapping packing of the items into square bins so that for all $j \in [d]$, the sum of the $j^{\textrm{th}}$ weight of items in each bin is at most 1. This is a natural problem arising in logistics, resource allocation, and scheduling. Despite being well studied in practice, surprisingly, approximation algorithms for this problem have rarely been explored. We first obtain two simple algorithms for GVBP having asymptotic approximation ratios $6(d+1)$ and $3(1 + \ln(d+1) + \varepsilon)$. We then extend the Round-and-Approx (R&A) framework [Bansal-Khan, SODA'14] to wider classes of algorithms, and show how it can be adapted to GVBP. Using more sophisticated techniques, we obtain better approximation algorithms for GVBP, and we get further improvement by combining them with the R&A framework. This gives us an asymptotic approximation ratio of $2(1+\ln((d+4)/2))+\varepsilon$ for GVBP, which improves to $2.919+\varepsilon$ for the special case of $d=1$. We obtain further improvement when the items are allowed to be rotated. We also present algorithms for a generalization of GVBP where the items are high dimensional cuboids

Tight Approximation Algorithms For Geometric Bin Packing with Skewed Items

In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. The problem admits no APTAS and the current best approximation ratio is 1.406 by Bansal and Khan [SODA\u2714]. A well-studied variant of the problem is Guillotine Two-dimensional Bin Packing (G2BP), where all rectangles must be packed in such a way that every rectangle in the packing can be obtained by recursively applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal, Lodi, and Sviridenko [FOCS\u2705] obtained an APTAS for this problem. Let ? be the smallest constant such that for every set I of items, the number of bins in the optimal solution to G2BP for I is upper bounded by ? opt(I) + c, where opt(I) is the number of bins in the optimal solution to 2BP for I and c is a constant. It is known that 4/3 ? ? ? 1.692. Bansal and Khan [SODA\u2714] conjectured that ? = 4/3. The conjecture, if true, will imply a (4/3+?)-approximation algorithm for 2BP. According to convention, for a given constant ? > 0, a rectangle is large if both its height and width are at least ?, and otherwise it is called skewed. We make progress towards the conjecture by showing ? = 4/3 for skewed instance, i.e., when all input rectangles are skewed. Even for this case, the previous best upper bound on ? was roughly 1.692. We also give an APTAS for 2BP for skewed instance, though general 2BP does not admit an APTAS

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

Online Two-Dimensional Load Balancing

In this paper, we consider the problem of assigning 2-dimensional vector jobs to identical machines online so to minimize the maximum load on any dimension of any machine. For arbitrary number of dimensions d, this problem is known as vector scheduling, and recent research has established the optimal competitive ratio as O((log d)/(log log d)) (Im et al. FOCS 2015, Azar et al. SODA 2018). But, these results do not shed light on the situation for small number of dimensions, particularly for d = 2 which is of practical interest. In this case, a trivial analysis shows that the classic list scheduling greedy algorithm has a competitive ratio of 3. We show the following improvements over this baseline in this paper: - We give an improved, and tight, analysis of the list scheduling algorithm establishing a competitive ratio of 8/3 for two dimensions. - If the value of opt is known, we improve the competitive ratio to 9/4 using a variant of the classic best fit algorithm for two dimensions. - For any fixed number of dimensions, we design an algorithm that is provably the best possible against a fractional optimum solution. This algorithm provides a proof of concept that we can simulate the optimal algorithm online up to the integrality gap of the natural LP relaxation of the problem

Online erőforrás allokációs problémák = Online resource allocation problems

Jellemző probléma, hogy korlátozott mennyiségű erőforrást kell szétosztani a felmerülő igények között. Számos esetben a probléma online, azaz a probléma inputját csak részenként ismerjük meg és döntéseinket a már megkapott információk alapján a további adatok ismerete nélkül kell meghoznunk. A kutatásaink során ilyen, online erőforrás allokációs problémákkal foglalkoztunk az alábbi részterületeken. Az ütemezés elméletében olyan problémákat vizsgáltunk, amelyekben az ütemezésen kívül más döntéseket is meg kell hozni az algoritmusnak, speciálisan a gépköltséges (a gépek száma nem adott paraméter, hanem meg kell őket vásárolni) és visszautasítható munkák modelljeit tanulmányoztuk. Ládapakolás és ládafedés esetén azon modelleket vizsgáltuk, amelyekben a ládák tartalmára a megkövetelt összsúlyon kívül további korlát is adott (pl a ládában szereplő elemek számára, vagy az elemek fajtáinak számára). Egy további modellt is vizsgáltunk, ahol a ládák száma adott és a cél a minimális összsúllyal lefedni az összes ládát. A nyugtázási probléma során előrenéző algoritmusokat vizsgáltunk, amelyek csak félig online algoritmusok, mert a döntés időpontjában az input egy további részéről (de nem az egészről) is rendelkeznek információval. A vizsgált modellek megoldására új algoritmusokat fejlesztettünk ki, azokat a versenyképességi elemzés alapján megvizsgáltuk. Néhány esetben, ahol a versenyképességi elemzés nem volt releváns empirikus elemzést hajtottunk végre. | In optimization problems it often happens that we have to allocate some bounded resources. In some cases these problems are online, which means that we receive the input part by part, and we have to make the decision without any information about the further parts. In the project we analysed such problems. In scheduling the we considered the problem of scheduling with machine cost (the number of machines is not part of the input, we have to buy it) and scheduling where the jobs can be rejected. In bin packing and covering we considered the problems where extra condition is given for the bins (the number of items, or the number of items of different types). Furthermore we investigated a model, where the number of bins is given, and the goal is to minimize the total size of items which cover them. In data acknowledgment we considered the time lookahead version, where the algorithm has some partial extra information about the further part of the input. In each model we developed new algorithms and analysed them by competitive analysis. In some cases where competitive analysis was not useful we used empirical analysis